Insolubles

The medieval name for paradoxes like the famous Liar Paradox
(“This proposition is false”) was “insolubles”
or insolubilia,
[1]
though besides semantic paradoxes, they included epistemic paradoxes,
e.g., “You do not know this proposition”. From the
late-twelfth century to the end of the Middle Ages and beyond, such
paradoxes were discussed at length by an enormous number of authors.
Yet, unlike twentieth century interest in the paradoxes, medieval
interest seems not to have been prompted by any sense of theoretical
“crisis”.

The history of the medieval discussions can be divided into three main
periods: (a) an early stage, from the late-twelfth century to the
1320s; (b) a period of especially intense and original work, during
roughly the second quarter of the fourteenth century; (c) a late
period, from about 1350 on. The discussion in this article will be
organized as follows:

1. Origins of the Medieval Discussion

The Liar Paradox was well known to antiquity. Its discovery is often
credited to Eubulides the Megarian (4th century BCE), on the basis of
a remark by Diogenes Laertius (Lives of the Philosophers
II.108), although in fact Diogenes says only that Eubulides discussed
the paradox, not that he discovered
it.[2]
A little later, the poet and grammarian Philetus (or Philitas) of Cos
(c. 330–c. 270 BCE), if we are to believe the story in Athenaeus
of Naucratis’s Deipnosophists IX.401e, worried so much
over the Liar that he wasted away and died of insomnia, as, according
to Athenaeus, his epitaph recorded:

Philetus of Cos am I
’Twas The Liar who made me die,
And the bad nights caused
thereby.[3]

Diogenes Laertius also reports (VII.196–98) that, in addition to
a huge number of other works on a variety of topics, the Stoic
logician Chrysippus (c. 279–206 BCE) wrote:

Introduction to the Liar;

Liar Propositions: An Introduction;

six books on the Liar itself;

Reply to Those Who Think There Are Propositions That Are Both
True and False;

Reply to Those Who Solve the Liar Proposition by
Division;

On the Solution to the Liar (in three books);

Reply to Those Who Say The Liar Argument Has False
Premises.

Unfortunately, almost none of Chrysippus’ work survives.

1.1 Unlikely Ancient Sources

Nonetheless, it does not appear that medieval interest in insolubles
was derived directly from these or any other known ancient sources
that discuss the Liar. Many of the relevant works were lost (e.g., the
works of Chrysippus), while others were never translated into Latin
and so were effectively unavailable to the Latin Middle Ages, though
things may be different for the Arabic and Byzantine traditions, which
are only starting to be studied (see, e.g., Alwishah & Sanson
2009; Gerogiorgakis 2009). Indeed, it is not at all clear just what it
was that prompted medieval interest. One might have supposed that,
even if particular theories about the Liar were not transmitted to the
Latin West from antiquity, at least formulations of Liar-type
paradoxes must have been known and available to stimulate the medieval
discussions. In fact, however, there are strikingly few
possibilities.

Seneca (Epistle 45.10), for instance, mentions the Liar
paradox by its Greek name pseudomenon, but does not actually
formulate it. Again, St Augustine perhaps has the Liar in mind in his
Against the Academicians (Contra academicos
III.13.29), where he refers to the “most lying calumny,
‘if it is true [it is] false, if it is false it is
true’”. But neither passage would likely be sufficient by
itself to suggest the special problems of the Liar to anyone not
already familiar with them.

Somewhat more explicit is Aulus Gellius’s (2nd century CE)
Attic Nights (XVIII.ii.10), “When I lie and say I am
lying, am I lying or speaking the truth?” But Gellius was not
widely read in the Middle Ages, and no known medieval author cites him
in the context of
insolubles.[4]
Again, Cicero’s Academica priora,
II.xxix.95–xxx.97, contains a fairly clear formulation:

If you lie and speak that truth [namely, that you lie], are you lying
or speaking the truth? … If you say you lie, and you speak the
truth, you lie; but you say you lie, and you speak the truth;
therefore, you lie.

But this passage is never cited in the
insolubilia-literature. Moreover Cicero, who wrote in Latin
and so did not have to be translated to be available to the Middle
Ages, calls such paradoxes “inexplicables”
(inexplicabilia). If he was the catalyst for the medieval
discussions, we would have expected to find that term in the
insolubilia-literature, and we do not; the unanimous medieval
term is ‘insolubles’.

1.2 St. Paul’s Reference to Epimenides

One initially plausible stimulus for the medieval discussions would
appear to be the Epistle to Titus 1:12: “One of themselves, even
a prophet of their own, said, The Cretians [= Cretans] are always
liars, evil beasts, slow bellies”. The Cretan in question is
traditionally said to have been the sixth-century BCE thinker
Epimenides. For this reason, the Liar Paradox is nowadays sometimes
referred to as the “Epimenides”. Yet, blatant as the
paradox is here, and authoritative as the Epistle was taken to be, not
a single medieval author is known to have discussed or even
acknowledged the logical and semantic problems this text poses. When
medieval authors discuss the passage at all, for instance in
Scriptural commentaries, they seem to be concerned only with why St.
Paul should be quoting pagan
sources.[5]
It is not known who was the first to link this text with the Liar
Paradox.

1.3 Aristotle’s Sophistical Refutations

By contrast with these passages, none of which was cited in the
insolubilia-literature, there is a text from
Aristotle’s Sophistical Refutations 25, [A-SR]:
180a27–b7, that, almost from the very beginning of the
insolubilia-literature to the end of the Middle Ages, served
as the framework for discussing insolubles. It occurs in
Aristotle’s discussion of the fallacy of confusing things said
“in a certain respect” (secundum quid) with
things said “absolutely” or “without
qualification” (simpliciter). In this context,
Aristotle considers someone who takes an oath that he will become an
oath-breaker, and then does so. Absolutely or without qualification,
Aristotle says, such a man is an oath-breaker, even though with
respect to the particular oath to become an oath-breaker he
is an oath-keeper. Then Aristotle adds the intriguing remark,
“The argument is similar too concerning the same man’s
lying and speaking the truth at the same time” ([A-SR]:
180b2–3). It was this sentence that many medieval authors took
to be a reference to the Liar Paradox, which therefore, on the
authority of Aristotle, could be solved as a fallacy secundum quid
et simpliciter.

The widespread appeal to this passage throughout the history of the
insolubilia-literature indicates that the text did play some
role in prompting medieval interest in insolubles. This suggestion is
reinforced by the fact that the earliest known medieval statement of
the Liar occurs in 1132, around the time the Sophistical
Refutations first began to circulate in Western Europe in Latin
translation (see
Section 2
below).

Nevertheless, it is not immediately obvious how Aristotle’s
remarks can be made to fit the Liar Paradox. The oath-breaker, as the
example was generally interpreted, takes two oaths: one,
which he keeps, that he will commit perjury, and a second (it does not
matter what it is) that he breaks, thereby fulfilling the first oath.
The man is an oath-breaker and an oath-fulfiller, but with respect to
different oaths; by breaking his second oath, rendering it
false, he fulfills the first oath, making it true. However, it is
possible to interpret the passage as referring to a single oath, when
the oath is broken at the same time as it is made. Seen that way, it
connects the Liar paradox with the fallacy secundum quid et
simpliciter.[6]

In short, it seems clear that the Sophistical Refutations was
instrumental in prompting medieval interest in insolubles. But more
must have been involved too. Martin (1993) suggests a connection with
theories of obligations (cf.
Section 3.3
below). Before medieval logicians could formulate genuine Liar-type
paradoxes, they first had to go well beyond anything found in
Aristotle’s text. At present we cannot say whether they did this
on the basis of some still unidentified ancient source or whether it
was through their own intellectual power and logical insight.

1.4 The Many Varieties of Insolubles

The medievals discussed many more insolubles than the Liar Paradox,
though most can be seen as variants of it. One common variant was what
is now called the ‘yes’-‘no’ paradox: Socrates
says “What Plato is saying is false”, while Plato says
“What Socrates is saying is true” (see, e.g., Buridan
[B-SD]: 974). There is also the ‘no’-‘no’
paradox, where Plato instead says “What Socrates is saying is
false” (Buridan [B-SD]: 971). There is even a
‘no’-‘no’-‘no’ paradox, where
Socrates says that Plato is saying something false, Plato that Cicero
is saying something false and Cicero that Socrates is saying something
false (Albert [AS-I]: 353). There are conjunctive insolubles, e.g.,
“God exists and some conjunction is false”, where God has
annihilated any other conjunction, and disjunctive insolubles, e.g.,
“A man is an ass or some disjunction is false”, where God
has instead annihilated any other disjunctive proposition (Albert
[AS-I]: 357–8). There is also a nice example where a landowner
has decreed that only those who speak truly will be allowed across his
bridge and those who lie about their business will be thrown in the
water (or maybe even hanged on the nearby gallows). When Socrates is
challenged on coming to the river, he says “You will throw me in
the water” (Bradwardine [B-I]: 135; Buridan [B-SD]: 993; see
also Cervantes Don Quixote, vol. II book III ch. XIX, p.
714).

The medievals discovered what is now usually referred to as
Curry’s Paradox, either in the form “If this conditional
is true then a man is an ass” (see Read 2009: section 9), or in
its contraposed form, “If God exists then some conditional is
false” (where this is the only conditional: Albert [AS-I]: 359).
There were also epistemic paradoxes, such as “You do not know
this proposition” (Bradwardine [B-I]: 139): you do not know it,
for if you did, it would be true and you would not know it. But now
you must realise that you do know it. Such insolubles can involve
doubt as well as knowledge, e.g., “Socrates knows the
proposition written on the wall to be doubtful to him” (see
note 32).
Further insolubles arise from the medieval theory of (logical)
Obligations,
e.g., “Something proposed is denied by you” (Bradwardine
[B-I]: 125), and “The king is sitting or a disjunctive doubt is
proposed” (Bradwardine [B-I]: 151)—in the theory of
obligations, the respondent is taken never to know whether the king is
sitting.

2. Early Developments to the 1320s

In 1132, Adam of Balsham, the founder of the important logical school
of the “Parvipontani” (so called because they
gathered at the Petit Pont in Paris), wrote an Art of
Discussing (Ars disserendi), in which he treats, among
other things, various kinds of yes/no questions, including
“whether he speaks truly who says he lies” and
“whether he who says nothing but that he lies is speaking the
truth”. ([AB]: 107.)

The importance of this passage should not be exaggerated. It is true
that it gives us the earliest known explicit medieval formulation of
the
Liar.[7]
But Adam makes no attempt to solve the paradox, does not say it was a
current topic of discussion in his day, and in fact does not even
indicate he recognized its paradoxicalness. He simply offers it as an
example of one kind of yes/no question.

It is not until later in the twelfth century that one finds an
explicit statement of the special problems raised by insolubles. In
his On the Natures of Things (De naturis rerum), of
unknown date but apparently well known by the end of the century,
Alexander Neckham ([N-NR]: 289)
says[8]:

Again, if Socrates says he lies, and says nothing else, he says some
proposition. Therefore, either a true one or a false one. Therefore,
if Socrates says only that he lies, he says what is true or what is
false.

But if (1) Socrates says only the proposition that Socrates lies, and
he says what is true, then it is true that Socrates lies. And
if it is true that Socrates lies, Socrates says what is false.
Therefore, if Socrates says only the proposition that Socrates lies,
and he says what is true, he says what is false.

But if (2) Socrates says only the proposition that Socrates lies, and
he says what is false, then it is false that Socrates says
what is false. And if it is false that Socrates says what is false,
Socrates does not say what is false. But if Socrates says only that he
lies, he says either what is true or what is false. Therefore, if
Socrates says he lies, he says what is true. Therefore, if Socrates
says only that he lies, and he says what is false, then he says what
is true.

But if Socrates says only that he lies, he says what is true or false.
Therefore, if Socrates says only that he lies, he says what is true
and says what is false.

Nevertheless, although it is clear that Neckham was fully aware of
what is paradoxical about the Liar, he makes no attempt to
solve the paradox. He presents it only as an example of the
“vanities” logic deals with. This suggests that by his day
others were trying to solve the paradox, and in fact in the
discussion of the fallacy secundum quid et simpliciter
contained in the so-called Munich Logic
(= Dialectica Monacensis) from sometime in the second
half of the century, we find the remark: “But how this fallacy
arises in uttering the insoluble ‘I am saying a
falsehood’, that is a matter discussed in the treatise on
insolubles”.[9]

The first text we have that actually tries to solve the paradox is an
anonymous treatise from the very end of the twelfth or the very early
thirteenth century (De Rijk 1966). From then on, there are a great
number of surviving treatments (see Spade 1975). In the early 1320s,
Thomas Bradwardine, in a preliminary section of his own treatise on
insolubles, lists nine views in circulation in his day, including his
own (see Bradwardine [B-I], ch. 2; Spade 1987: 43–46). Some of
these views can no longer be identified in the texts that survive from
the period before Bradwardine, but among the surviving views, we can
distinguish five broad approaches to “solving” the
paradox.[10]
(Sometimes these approaches are combined in a single author.)

2.1 Insolubles as Fallacies secundum quid et simpliciter

As might be expected in view of
Section 1.3
above, many of these early theories attempted to analyze insolubles
as fallacies secundum quid et simpliciter. Later in the
insolubilia-literature, discussions often continued to be
cast in terms of this fallacy. It was claimed in (Spade 1987: 32) that
their real focus was generally on entirely different theoretical
issues, and that the role of the fallacy thus became purely
“honorary”, preserving the authority of Aristotle.
However, that assessment was challenged in (Dutilh Novaes & Read
2008). Bradwardine, for example, explicitly and repeatedly casts his
solution in terms of Aristotle’s discussion, making actual (not
merely “honorary”) use of the conceptual framework offered
by the fallacy secundum quid et
simpliciter.[11]

Aristotle had suggested ([A-SR]: 180b5–7) that insolubles are
false simpliciter (absolutely/without qualification), but
true secundum quid (in a certain respect). Some authors in
the early medieval literature, however, argued that insolubles are
without qualification neither true nor false, but only true in a
certain respect and false in a certain
respect.[12]
Others used the terminology of simpliciter and secundum
quid, but applied it to reference (suppositio) rather
than to truth and falsehood, so that in insolubles certain terms did
not refer “without qualification” to their referents, but
only “in a certain respect”. This view is in effect a kind
of restriction on
self-reference.[13]

2.2 The Theory of Transcasus

The theory of transcasus has nothing to do with the fallacy
secundum quid et simpliciter, although it too seems to have
had its origins in antiquity. The word transcasus is not a
common Latin word. It seems to be a literal translation of Greek
metaptosis. In Stoic logic, propositions that change their
truth value over time were called metapiptonta (from the same
root). Walter Burley in fact used the word transcasus exactly
this way in 1302 in two short logical works (Spade 1987:
33–34).

Nevertheless, in the particular context of insolubles, while the term
transcasus does have an association with time, it does not
imply any change of truth value over time. Rather the theory
of transcasus held that in the proposition “This
statement is false”, the term “false” refers
not to the proposition in which it occurs, but rather to some
proposition uttered earlier. Thus, when the liar says “I am
lying”, what he really means is “What I said just a moment
ago was a lie”. If the speaker did not in fact say anything
earlier, then his present statement is simply false and no paradox
arises.[14]

This odd view, like the last of those discussed in
Section 2.1
above, amounts in practice to a restriction on self-reference. But it
is not clear exactly what motivated it. In any event, the theory of
transcasus appears to have disappeared as a theory actually
held by anyone after the early period, although it continued to be
mentioned in later authors’ surveys of earlier
views.[15]

2.3 Exercised Act vs. Signified (or Conceived) Act

A third theory from this early period distinguishes the
“exercised” act from the “signified” or
“conceived” act. The details of this theory are not yet
well understood, but the basic strategy is to distinguish what the
liar says he is doing (namely, lying) from what he really
is doing. The author of the Questions on the Sophistical
Refutations attributed to John Duns Scotus, who held a version of
this theory (Scotus [DS-Q], questions 52–53, pp. 505–15),
thought that what the liar is really doing (the “exercised
act”) is speaking the truth. In order to avoid the paradox, this
theory would seem to be committed to saying that the exercised act and
the signified act are two distinct acts, so that the theory,
like the theory of transcasus
(Section 2.2
above), is committed to some kind of restriction on
self-reference.[16]

2.4 The Theory of Restriction

Even when not combined with transcasus or the theory that
distinguishes exercised act from signified act, a very popular
approach throughout the insolubilia-literature, in the early
period (and for that matter even in our own day), before
Bradwardine’s sustained attack on it (see
Section 3.1),
was to deny or restrict the possibility of self-reference. Such
theories had the title “restriction” (restrictio)
and their proponents were called “restricters”
(restringentes). All such theories maintained that in some or
even all cases, terms in propositions could not “supposit
for” (stand for, refer to) the propositions in which they
occur.

Some theories of restriction went further and also ruled out other
patterns of reference. For example:

Proposition a = ‘b is true’, and
b = ‘a is false’ (the
‘yes’-‘no’ paradox). Here a refers to
b and b refers back to a. But reference is not a
transitive relation, so that there is no real self-reference
here. Nevertheless, the situation is paradoxical, and as a result some
authors ruled out all referential “loops”.

Proposition a is a certain token of the form
‘a is false’, and b is a second token of the
same type. Token a is self-referential, but token b is
not, since it refers to a, not to itself. Yet some authors
thought the two tokens should be treated semantically alike, so that
not only could the subject of a not refer to a itself,
neither could the subject of b.

Proposition a = ‘b is true’, and
proposition b = ‘b is false’. Here, b
is self-referential, but a is not. Nevertheless, b is
the contradictory of a. Hence, by saying its contradictory is
true, a is in effect saying that it itself is false. Thus,
although it is not self-referential, a is nevertheless
paradoxical. Some authors prevented such cases by maintaining that not
only were terms unable to refer to the propositions in which they
occurred, they also could not refer to the contradictories of
the propositions in which they occurred.

As a general theory, restriction is open to an obvious objection: it
rules out innocuous forms of reference along with pathological ones.
The sentence “This sentence has five words” is not
paradoxical, after all, even though it is self-referential; in fact,
it seems obviously true. Yet the general theory of restriction would
disallow it.

Medieval authors sometimes raised this objection. As a result, we find
two kinds of restriction-theories in the medieval literature:
(a) general or strong theories that rule out self-reference, and
perhaps other patterns of reference too, in innocuous as well as
pathological cases; and (b) more specialized or weaker theories that
rule out certain forms of reference only when they result in paradox.
Walter Burley and William of Ockham, for example, held the latter form
of restriction (Spade 1974).

If general or strong theories of restriction are open to the objection
stated above, the weaker theories are open to a different objection:
they risk being vacuous if their proponents did not have any
independent way of identifying paradoxical cases. Perhaps, their
theories amounted to saying “all forms of reference are allowed,
except for paradoxical ones, which are not allowed”. This is no
doubt true, but it is also a
tautology.[17]

The restrictivist response largely died out after Bradwardine’s
attack on it in chs. 3–4 of his Insolubles. However, in
the late 1320s, Walter Segrave (or Sexgrave) gave a spirited defence
of restrictivism against Bradwardine’s arguments (see Spade 1975
item LXVIII, pp. 113–6).

2.5 Cassation

Unlike restriction, which remained (and remains) a popular view, the
theory of “cassation” disappeared very early (though it
has its contemporary advocates). It is maintained in the earliest
known treatise on insolubles (De Rijk 1966) and in one other early
anonymous text (Spade 1975: 43–44), but seems to have died out
after about 1225, although it continued to be mentioned in later
authors’ surveys of previous views, no doubt because of its
inclusion in Bradwardine’s own survey. It was briefly revived by
John Dumbleton in the 1330s: see (Spade 1975 item XXXVI, pp.
63–5). His essential idea was that signification requires
uptake, so any utterance which cannot be understood cannot constitute
a proposition—and the insolubles defy understanding, for
self-reference generates a regress of deferred intelligibility.

‘Cassation’ is now an archaic word (though it survives in
legal documents), but merely means “making null and void,
canceling”. In effect, this theory holds that one who utters an
insoluble proposition “isn’t saying anything”. The
second of the texts just cited even gives a curious “ordinary
language” argument, appealing to the rusticus (the
man-on-the-street), who, if you were to say to him “What I am
saying is false”, would reply “Nil dicis”
(“You are saying nothing”).

The treatise in De Rijk 1966 presents more of a theory. Much of it is
obscure to modern scholars, but it seems to appeal to a distinction
between a mental act of asserting and a vocal act of uttering a
proposition. “Saying” requires both acts; it is “an
assertion with utterance”. In the case of the liar who says
“What I am saying is false”, the mental act of asserting
is present, and for that matter so is the physical act of uttering the
words. But somehow (this is the obscure part) there is no
“saying”.

It is tempting to interpret this view as an appeal to a kind of
fallacy of composition; just as someone who is both good and an author
is not necessarily a good author, so too something that is both
mentally asserted and vocally uttered is not necessarily
“said” (asserted with utterance). It is tempting, yes, but
highly speculative. Nevertheless, whatever the correct interpretation,
it appears that the distinction between asserting and uttering drawn
by this theory escapes the facile “refutation” of it used
as early as the mid-thirteenth century, that it “plainly
contradicts sensation that is not
deceived”.[18]

3. The Second Quarter of the Fourteenth Century

The preceding theories represent the earliest stage of the
insolubilia-literature. Although these theories are sometimes
mentioned in the later literature, and in the case of
“restriction” often accepted in the later
literature, much more sophisticated treatments began to emerge in the
second quarter of the fourteenth century. The turning point is Thomas
Bradwardine, whose own theory was enormously influential on later
authors. Shortly after Bradwardine, two other English authors from
this middle period are also important: Roger Swyneshed
(Section 3.2),
and William Heytesbury
(Section 3.3).
A little later, important contributions were made by Parisian
authors, Gregory of Rimini
(Section 3.4),
John Buridan
(Section 3.5)
and Albert of Saxony
(Section 3.6).

3.1 Thomas Bradwardine

Thomas Bradwardine (c. 1300–1349) wrote his Insolubles
at Oxford sometime between 1321 and 1324. It became one of the most
important works on the topic in the Middle Ages. In fact, early in the
third quarter of the fourteenth century, Ralph Strode, in his own
treatise on the topic, surveys the earlier views (quoting
Bradwardine’s own survey and theory almost verbatim), and then
says (Spade 1981: 116):

For the opinions mentioned above were those of the old [logicians],
who understood little or nothing about insolubles. After them there
arose the prince of modern philosophers of nature, namely Master
Thomas Bradwardine. He was the first one who discovered something
worthwhile about insolubles.

Bradwardine’s theory is built around a distinctive theory of
truth, which in turn depends on a conception of signification,
described by Spade as an “adverbial” theory of
propositional signification (Spade 1996: 178–85
[Other Internet Resources];
cf. Read 2008b: §13.2). By virtue of their constituent terms,
propositions signify things; but, in addition, a proposition as a
whole signifies that such-and-such is the case. This
conception may be related to Walter Burley’s theory of the
propositio in re (see, e.g., Cesalli 2001). It is this latter
kind of signification that is the basis for Bradwardine’s theory
of truth.

For Bradwardine, a proposition is (D1) true if it signifies
only as is the case (tantum sicut est), and (D2)
false if it signifies otherwise than is the case (aliter quam
est). Note the absence of the ‘only’ in the criterion
for falsehood. Truth therefore, is more demanding than falsehood. In
order for a proposition to be true, all of what it signifies
to be the case must in fact be the case; if any of what it
signifies to be the case fails to be the case, the
proposition is false. He will then argue that insolubles signify more
than at first appears, and that not everything they signify can be the
case. Consequently, they are simply false.

Thus what is most distinctive in Bradwardine’s theory is his
“multiple-meanings” theory of signification. For him,
propositions signify many things, not in the sense of being ambiguous,
but as requiring a multitude of conditions to be satisfied for their
truth. For example, ‘Some man is running’ signifies not
only that a man is running, but also that there is a man, and a
runner. Indeed, Bradwardine claims that a proposition signifies
everything which follows from it. This is his famous second postulate,
(P2). There is considerable controversy over its correct
interpretation (for a careful discussion, see Dutilh Novaes 2009:
§1). (P2) is interpreted by Spade (1981:120) as what he calls
“Bradwardine’s Principle” (BP):

If p only if q, then P signifies that
q,

where the name replacing ‘P’ names the sentence
replacing ‘p’. However, he concedes that, when read
in this way, the principle does not support the proof Bradwardine
gives of his second thesis, (T2), which we will discuss below.
Accordingly, Spade attributes a further principle to Bradwardine, the
“Converse Bradwardine Principle”
(CBP):[19]

Whatever a sentence signifies follows from it. If P signifies
that q, then p only if q.

However, he admits that Bradwardine never states or mentions this
principle, and that with it, Bradwardine’s solution
collapses.

It is claimed in Read 2009 that (P2) should be interpreted more
generously, not in terms of how it is actually stated by Bradwardine,
but how it is actually used by him. As used, it is a closure
principle, that a proposition signifies everything which follows from
what it signifies. This arguably has (BP) as a consequence, but is
stronger than it, and sufficient for Bradwardine’s proof of
(T2).

Bradwardine’s solution to the insolubles is stated in his second
thesis, (T2): “Every proposition which signifies that it itself
is not true, or is false, also signifies that it is true and is
false”. The proof has four stages:

suppose first that a signifies that a is not true,
and nothing else. If a is not true, then by (D1) it does not
signify only as is the case, so it is not the case that a is
not true (since we are supposing that is all it signifies), that is,
a is true. So if a is not true, it is true. But a
signifies that a is not true, so by (P2) a signifies
that it is true. Thus a does not and cannot signify only that
a is not true.

So suppose that a signifies that a is not true and
also that b is c. If a is not true, then by (D1)
it does not signify only as is the case, so it is not the case that
a is not true and b is c, that is, either
a is true or b is not c, by (P4), a statement of
the De Morgan Laws. So again by (P2), a signifies either that
a is true or b is not c. But a signifies
that b is c, so by (P5), Disjunctive Syllogism, and (P2)
again, a signifies that a is true.

suppose that a signifies that a is false. Then by
(P1), Bivalence, and (P2), a signifies that a is not
true, so by (1) and (2) above, a signifies that a is
true.

so if a signifies that a is not true or that
a is false, a also signifies that a is true. But
a cannot be both true and false. So things cannot be only as
a signifies, so by (D2) a is false.

In the subsequent chapter, Ch. 7, Bradwardine considers the problem of
revenge in various guises. (See, e.g., the entry on the
Liar paradox, and especially
the section on Expressive power and ‘revenge’.)
Take Socrates’ utterance of
‘Socrates utters a falsehood’, where Socrates utters
nothing else. Bradwardine’s claim is that Socrates’
utterance is false, that is, that Socrates utters a falsehood. How can
Bradwardine’s claim that Socrates utters a falsehood be true,
while Socrates’ utterance of the same thing is false? The
reason, Bradwardine replies, is that Socrates’ utterance is
self-referential, and signifies not only that Socrates’
utterance is false but also, by (T2), that it is true (and so is
false), whereas Bradwardine’s utterance is not self-referential
and so not subject to
(T2).[20]

3.2 Roger Swyneshed

Sometime between roughly 1330 and 1335, the English Benedictine Roger
Swyneshed adopted a theory in some respects reminiscent of
Bradwardine’s, but with interesting features of its own. Like
Bradwardine, Swyneshed held that for a proposition to be true, it is
not enough that it “signify as is the case”. But whereas
Bradwardine maintained that in addition the proposition must not
signify otherwise than is the case (that is, it must signify
only as is the case), Swyneshed said that in addition the
proposition must not “falsify itself”. Insolubles do
falsify themselves, and so are false for that reason, even though they
signify as is the case. Propositions that falsify themselves are said
to be those that are “relevant (pertinens) to inferring
that they are false”.

The notions of “relevance”,
“self-falsification”, and “signifying as is the
case” (or “otherwise than is the case”) are puzzling
ones in Swyneshed’s theory and the subject of ongoing
study.[21]
But the main historical interest of his theory does not lie there.
Rather, it lies in three famous and controversial conclusions he drew
from his principles:

Some false propositions signify as is the case. Insolubles
do.[22]
Thus, where a is the insoluble ‘a is
false’, a is self-falsifying and so false. But it
signifies as is the case (namely, that it is false).

In some valid formal inferences, falsehoods follow from truths.
For consider the inference “The conclusion of this inference is
false; therefore, the conclusion of this inference is false”.
The premise and the conclusion of this inference are two tokens of the
same type, so, Swyneshed claimed, the inference is a formally valid
one, an instance of simple repetition. (Bradwardine and Buridan would
both disagree.) But while the conclusion is a self-falsifying
insoluble, and so is false, the premise is not
self-falsifying, and is in fact true. (The conclusion of the
inference is false, on Swyneshed’s account.) Here then,
a falsehood validly follows from a truth.

In the case of insolubles, two mutually contradictory propositions
are false at the same time. Where a = ‘a is
false’, a is insoluble and false. But its contradictory,
‘a is not false’, Swyneshed claims, is
not insoluble and is not self-falsifying. Nevertheless, it is
false because it signifies otherwise than is the case. The insoluble
a really is
false.[23]

Many authors found these conclusions ridiculous, especially the second
and third ones. But they had their defenders as
well.[24]

Two other features of Swyneshed’s theory should be at least
mentioned, although our understanding of his view does not yet allow a
thorough treatment of them. Along with other authors (e.g., Buridan),
he explicitly holds that while valid inference does not always
preserve truth, it does preserve the property of signifying as is the
case. Second, Swyneshed explicitly considers a situation where
a = ‘a does not signify as is the case’, and
says that a is neither true nor false in that situation. This
is the only known case of a medieval author’s actually allowing
failure of bivalence for insolubles, even though several authors refer
to (and reject) such
theories.[25]

3.3 William Heytesbury

In 1335, the Mertonian logician and philosopher of nature William
Heytesbury wrote an important treatise Rules for Solving
Sophisms (Regulae solvendi
sophismata).[26]
The first of its six chapters is on insolubles. The Rules as
a whole, and this first chapter in particular, were widely read and
commented on, particularly in Italy in the late-fourteenth and
fifteenth centuries (see, e.g.,
Section 4.3
below). Indeed, Heytesbury’s theory is a competitor to
Bradwardine’s as the most influential theory of insolubles in
the whole of the Middle
Ages.[27]

Heytesbury treated insolubles as paradoxical only with respect
to certain assumed circumstances (what he calls the
casus or “hypothesis”). For example, the
proposition ‘Socrates is uttering a falsehood’ is not
paradoxical in the abstract, all by itself, but only in contexts
where, say, it is Socrates who utters that proposition, the
proposition is the only proposition Socrates utters (it is
not an embedded quotation, for instance, part of some larger statement
he is making), and where his proposition signifies just as it normally
does. Spoken and written language are thoroughly arbitrary, for
medieval authors, so that the vocal sequence or inscription
‘Socrates is uttering a falsehood’ could theoretically
signify any way you want. It might, for example, signify that 2 + 2 =
4, in which case it would not be insoluble at all but
straightforwardly true.

It is the last condition that is the focal point for
Heytesbury’s attack. He holds that in the casus where
Socrates himself says just ‘Socrates is uttering a
falsehood’ and nothing else, his proposition cannot, on
pain of contradiction, signify just as it normally does
(precise sicut verba communiter pretendunt, as he puts it).
If it does signify as it normally does, it must signify some other way
as well.

How else might it signify? Heytesbury did not think it was his duty to
answer that question, as Bradwardine did. The proposition’s
additional signification cannot be predicted, given the arbitrariness
of spoken and written language. Depending on what else it signifies,
different verdicts about the proposition are appropriate. In short,
Heytesbury’s strategy is to say,

You tell me exactly what Socrates’s statement
signifies, and I’ll tell you first of all whether the case you
describe is possible, and if it is, I’ll tell you whether his
statement is true or false.

This “shift the burden” strategy is a consequence of the
fact that Heytesbury, even more than Bradwardine and Swyneshed, views
the question of insolubles in the context of obligationes, a
highly formalized medieval disputation context that is a subject of
much recent
discussion.[28]
But many later authors felt that Heytesbury had simply sidestepped
the real theoretical issue, and went on to stipulate what Heytesbury
would not: an insoluble’s “additional”
signification. They held that, in circumstances that make it
insoluble, a proposition not only signifies as it normally does; it
also signifies that it is true. This “adjustment”
to Heytesbury’s theory has the effect of combining it with the
tradition stemming from
Bradwardine.[29]
It proved to be an appealing combination.

3.4 Gregory of Rimini

Gregory of Rimini’s main writing was done in the 1340s. Although
today we know of no text or passage of his that discusses insolubles,
there must have been one, because in 1372 Peter of Ailly cites
Gregory’s theory in some detail and uses it in writing his own
treatise on insolubles (see Peter of Ailly [P-CI], and
Section 4.2
below).

Gregory’s view relied on the traditional medieval notion (going
back to Aristotle’s On Interpretation 1, 16a3–5)
of “mental language”, the “language of
thought” that underlies and is expressed in spoken and written
language.[30]
Unlike spoken and written languages, where the signification of words
and propositions is thoroughly a matter of convention, signification
in mental language is fixed by nature once and for all, the same for
everyone. It follows that propositions in mental language can never
signify otherwise than they “normally” do. Thus
Heytesbury’s analysis, according to which insolubles do signify
otherwise than they normally do, cannot be applied to propositions
formed in mental language. Although Heytesbury himself did not draw
this conclusion, it follows from his theory that insolubles cannot be
formulated in mental language.

In the absence of any text by Gregory on the topic, we cannot be sure
that he reasoned like this from Heytesbury’s position. But for
whatever reason, he apparently did confine insolubles to spoken and
written language; for Gregory there are no insolubles in mental
language. An insoluble proposition in spoken or written language
corresponds to and expresses not the mental proposition one
would normally expect on the basis of the usual linguistic
conventions, but a complex and non-paradoxical mental
proposition.

For example, where a is the spoken or written proposition
‘a is false’, a corresponds to and expresses
the conjunction of two mental propositions. The first
conjunct signifies that a is false. Note that this is not the
insoluble a, since that was in spoken or written language
whereas this proposition is mental. Unlike a, this proposition
is not self-referential; it refers instead to a.

The second conjunct signifies that the first conjunct is
false. Since the first conjunct signifies that a is false, this
means that the second conjunct amounts to saying that a is
not false, but rather true.

One way, therefore, of viewing Gregory’s theory is to say that
he adopted the hybrid view described at the end of
Section 3.3
above, the view that combines Heytesbury with Bradwardine, but then
moved that whole analysis into mental language. Just as for
Heytesbury’s theory, insolubles for Gregory do not signify just
as their words normally do—they do not express the mental
proposition one would expect from the normal linguistic conventions).
Just as for Bradwardine’s theory, insolubles for Gregory do
signify partly that way (through the first conjunct of the mental
proposition). But they do not signify precisely that way;
they also signify that they are true (through the second conjunct of
the mental proposition).

Given our present knowledge of Gregory’s views, this
reconstruction must remain speculative.

3.5 John Buridan

John Buridan was another logician of the 14th century also
holding a theory of the insolubles similar to Bradwardine’s.
However, there may have been no direct influence. Buridan taught at
the University of Paris, and rather unusually remained a teaching
master in the Arts Faculty for his entire career, from the 1320s until
at least 1358. We can trace the development of his approach to the
insolubles from his early Treatise on Consequences, dating
from the 1330s, through his commentaries on Aristotle’s
Posterior Analytics, Sophistical Refutations, and
Metaphysics, to the various treatises of the Summulae de
Dialectica, which was repeatedly revised over twenty years. His
final view is described in the ninth and last treatise of the
Summulae, with the independent title Sophismata, in
a version from the mid-1350s (see Pironet 1993).

Buridan’s early view was that every proposition, not just
insolubles, signifies its own truth. This idea can be found as early
as Bonaventure’s Quaestiones disputatae de mysteria
Trinitate (q1 a1), composed in the 1250s. Unlike Bradwardine,
Buridan gives only the briefest of arguments for this claim about
signification, not grounding it in any principle like
Bradwardine’s (P2). Thus insolubles, which signify their own
falsity, signify that they are both true and false, and so are false.
Buridan’s view on insolubles may have developed out of Girard of
Odo’s, whose Logica was composed in Paris in the early
1300s. Girard’s idea was that the Liar paradox, ‘I say
something false’, has four things wrong with it
(malitiae), the fourth being that, since it is affirmative,
it asserts the unity of its subject with its predicate, but its
predicate (‘something false’) denies this (see Giraldus
Odonis Logica, 396–8).

However, even in the Treatise on Consequences, Buridan thinks
this final step, concluding that insolubles are false because they
signify that they are both true and false, needs qualification. For he
rejects the idea that a proposition is true if things are how it
signifies, even however it signifies. Things can be, e.g., how
‘No proposition is negative’ signifies, but it cannot be
true (since it is itself negative, and thus falsifies itself when
formed). Rather, affirmative propositions are true if their terms
supposit for the same, negative if they supposit for different things.
Indeed, he later rejects the suggestion that propositions signify
their own truth. He did so both for ontological reasons, since that
would require some kind of propositional meanings (the famous
complexe significabilia—see, e.g., Klima 2009:
§10.2); and also because that would make every proposition
meta-linguistic. Rather, his later theory claims that every
proposition virtually implies another proposition asserting the truth
of the first. Then an insoluble like ‘Socrates utters a
falsehood’, uttered by Socrates, is false not because its terms
don’t supposit for the same, but because the terms in the
implied proposition can’t also do so. Buridan’s solution
has been much discussed in recent decades and has been edited and
translated several times (see Buridan [B-S], [B-S2], [B-B], [B-SD],
[B-SD2]), but it is deeply problematic (see, e.g., Read 2002: §5;
Read 2006: §6).

3.6 Albert of Saxony

Much closer to Bradwardine’s theory than to Buridan’s is
that of another Paris logician, Albert of Saxony, who arrived in Paris
sometime before 1351 and taught there until around 1362. His view of
the insolubles is similar to Buridan’s early view, arguing in a
similar way that every proposition signifies its own truth. But there
is reason to doubt whether Albert was a student or even follower of
Buridan, for Buridan belonged to the Picardian Nation at the
University, whereas Albert was in the English (or by then
“Anglo-German”) Nation, and in general his outlook follows
the English logical tradition from earlier in the century. What is
perhaps most impressive, and enjoyable, about Albert’s treatise
on Insolubles (Albert of Saxony, [AS-I]), the first part of
the sixth treatise of his Perutilis Logica (A Really
Useful Logic), is the extensive list of insolubles treated, and
their variety. For example, we find there a paradox much discussed in
the recent literature on paradoxes under the title
‘V-Curry’, closely related to
Curry’s paradox.
In fact, in its contemporary form it appears in Dumbleton’s
discussion of the insolubles (see above,
Section 2.5)
and in Heytesbury’s Sophismata Asinina ([H-SA}:
sophism 18, p. 413): consider the consequence with sole premise
‘This consequence is valid’ and conclusion ‘A man is
an ass’. Albert contraposes it, inferring ‘This
consequence is invalid’ from the premise ‘God
exists’. The text translated in “Insolubles”
([AS-I]: XIV, p. 368) reads (slightly amended):

Let this consequence be A, its antecedent [‘God
exists’] B, and its consequent [‘This consequence
is invalid’] C, and let ‘this’ indicate the
consequence itself. Then I put forward consequence A, and I ask
whether consequence A is valid or not.

[i] If one says that it is valid, then, since its antecedent is true,
it follows that its consequent is true. And if its consequent is true,
then things are as its consequent signifies. But the consequent
signifies that consequence A is not valid. Therefore,
consequence A is not valid.

[ii] But if one says that consequence A is not valid—on
the contrary: If consequence A is not valid, it is possible
that B be true while C is false. But that is false,
which I prove as follows. For if A is not valid, things are as
C signifies them to be, because C signifies that
A is not valid; and consequently, C is true. Therefore
B cannot be true unless C is true. Therefore, if
A is not valid, A is valid. The first consequence is
evident, for in order that A be not valid it is enough that
B can be [true] without C, if they are formulated. The
last consequence holds “from the first to the last”.

The translation follows the Venice 1522 edition. As given in [AS-L]
(p. 1158), following the manuscripts, the text given in [ii]
reads:

[ii′] But if one says that consequence A is not
valid—on the contrary: If consequence A is not valid, it
is possible that B be true while C is false. But that is
false, which I prove as follows. For if A is not valid, then
C is true, and so A is valid, since B is not true
unless C is true. So if A is not valid, A is
valid. The premise [‘If A is not valid then C is
true’] is clear, for if A is not valid, things are as
C signifies, because C signifies that A is not
valid; and, consequently, C is true. The first consequence is
evident, for in order that A be valid it is enough that
B cannot be true without C, if it is formulated. The
last consequence holds “from the first to the last”.

In a nutshell, Albert’s argument in [ii′] is that if
A is invalid, it must be possible for B to be true and
C false. But if A is invalid, C is true. So it is
impossible for B to be true and C false. So even if
A is invalid, it is valid, and so A is valid. But by
[i], if A is valid it is invalid. So it is
both—paradox.

Dumbleton’s response to the paradox is to deny that the premise
(of his version, ‘This consequence is valid’) constitutes
a proposition, given the regressive reference of its subject
(‘this consequence’—which
consequence?—‘this consequence’, …; cf. Ryle
1951). Albert’s response, however, in line with his general
approach to the insolubles, is to agree that his inverted consequence
is indeed invalid, but to deny that it follows that C
(‘This consequence is invalid’) is true, for C
signifies more than just that A is invalid. Thus the premise
(‘God exists’) can be true without the conclusion being
true, and so the consequence really is invalid.

4. The Late Period

The period of greatest innovation and sophistication in the medieval
insolubilia-literature seems to have been the second quarter
of the fourteenth century. After about 1350, less original work is
known. Insolubles continued to be discussed, but it seems that for the
most part the theories adopted were variations or elaborations of the
ones already seen. Paul of Venice (1499), writing in about
1396–7, in the final section of his Logica Magna, lists
fifteen theories, supplementing Bradwardine’s list of nine with
later developments, but most from before 1350 (see Spade 1973:
82–4). This period is not yet well researched, however, so it is
too early for a clear verdict.

4.1 John Wyclif

One of the main (and perhaps genuinely new) theories to emerge from
this late period is that of John Wyclif, who wrote a Summa of
Insolubles (Summa insolubilium),
[31]
probably in the early 1360s, and included another discussion of
insolubles in his Continuation of the Logic (Logicae
continuatio), III.8. The theory is essentially the same in the
two treatments.

For Wyclif, the key to resolving insolubles is to recognize various
senses in which propositions can be true or false. There are three
main senses of ‘true’, and accordingly of
‘false’:

In the transcendental sense, truth is convertible with being, so
that any proposition is true in this sense, no matter what it
signifies. This sense can be disregarded in discussing insolubles.
Nothing (that is, no being) is false in the sense that it fails to be
true in this sense.

In a second sense, a proposition is true if and only if what it
“primarily signifies” exists. These “primary
significates” are neither substances nor accidents, but rather
“beings of reason”. It is perhaps plausible to interpret
an existing primary significate as analogous to a “fact”
in the modern philosophical sense. A proposition is false in this
second sense if and only if its primary significate fails to
exist.

In a third sense, a proposition is true if and only if what it
primarily signifies exists and is independent of the
proposition itself. It is false in this third sense if and only if its
primary significate either fails to exist or else exists but depends
on the proposition itself.

The “independence” required by the third kind of truth is
an obscure and difficult matter, not yet well understood. But here is
how it applies to insolubles:

Where a = ‘a is false’, its primary
significate either exists or does not. If it does, then in any event
it is not independent of a in the sense required by the third
kind of truth. In either case, then, a will be false
in the third sense. If the word ‘false’ in a is
taken in the second sense, therefore, a’s primary
significate does exist, since it is a fact that a is false in
the third sense. In short, the insoluble is true in the second sense,
but false in the third sense.

Our present understanding of Wyclif’s theory does not go much
beyond this. Many questions and problems remain. For instance, if the
word ‘false’ in a is not taken in the third sense
but in the second, the paradox seems to emerge all over again
in a form that cannot be handled by this theory.

Whatever its virtues or defects, Wyclif’s theory had some
influence on later authors. Robert Alyngton’s own
Insolubilia, for instance, from around 1380, explicitly
appeals to Wyclif’s theory. Its influence can also be seen in
Peter of Mantua’s account (see
Section 4.3
below) and in an anonymous late treatise preserved in a Prague
manuscript (see Wyclif [W-SI]: xxiv–xxv.)

4.2 Peter of Ailly

As already mentioned
(Section 3.4
above), in 1372 the Frenchman Peter of Ailly (Petrus de Alliaco)
wrote an Insolubilia that preserves all we know of Gregory of
Rimini’s theory. Peter’s theory looks much like
Gregory’s. Nevertheless, he did not accept Gregory’s view
entirely. Whereas for Gregory, an insoluble in spoken or written
language corresponds to or expresses a conjunction of two
propositions in mental language, for Peter it corresponds to or
expresses two distinct mental propositions, not their
conjunction. (The two distinct mental propositions are the same two
that Gregory had conjoined.)

In medieval semantics, propositions that correspond to two distinct
mental propositions are ambiguous or equivocal. (Indeed, that is the
medieval account of equivocation.) Thus, for Peter,
insolubles in spoken or written language are strictly equivocal and do
not have a single signification. In one sense (answering to the first
of Gregory’s conjuncts), they are true; in another sense
(answering to Gregory’s second conjunct), they are false. By
contrast, for Gregory, insolubles are just false, not ambiguous at
all; they correspond to a single false conjunction, one
conjunct of which is true and the other false.

Peter’s theory has the phenomenological advantage that it
accounts for the psychological “flip-flop” sense we have
when thinking about insolubles. When we look at them one way they seem
true; when we look at them another way, they seem false. No other
medieval theory seems to account for this psychological fact. For
further discussion, see Dutilh Novaes 2008a: §3.8.

4.3 Peter of Mantua

Strobino 2012 contains the first significant discussion of Peter of
Mantua’s account of insolubles in modern times. Mantua’s
treatise, composed in the early 1390s, shows influences from Albert of
Saxony and William Heytesbury (whose opinions he criticises
extensively) and of Wyclif’s theory. Mantua’s theory is
also mentioned in Paul of Venice’s Logica Magna, in
addition to the fifteen theories Paul specifically enumerates. Once
again, the theory frames itself in Aristotelian terms, whereby the
insolubles are absolutely or unqualifiedly false, but true in a
certain respect. Like Wyclif’s, his main notion of truth
(Wyclif’s third, Mantua’s second) requires that
propositions not be self-referential. If they are self-referential,
they are false in this sense. But Mantua’s other notion (similar
to but narrower than Wyclif’s second) applies only to
self-referential propositions that are true according to their primary
signification. For example, ‘This proposition is not true’
is false in the first sense (since it is self-referential), but true
in the second sense, because it is self-referential and not true (in
the first sense). Strobino argues that Mantua’s theory, like
Wyclif’s, cannot deal with the problem of revenge, e.g., with
such a proposition as ‘This proposition is not true in either
sense’.

5. Observations

Several instructive observations can be made about the medieval
insolubilia-literature.

First, although this article has focused on Liar-type paradoxes, and
although the medieval literature did too, it also included other kinds
of puzzles. For example, (the ‘no’-‘no’
paradox), where a = ‘b is false’ and
b = ‘a is false’, no Liar-type paradox
arises; contradiction can be avoided by simply taking one of the two
propositions as true and the other as false. But medieval logicians
regarded such cases as problematic because they require us to assign
different truth values to propositions that are semantically exactly
alike; there is no reason to pick a as the true proposition
rather than b or conversely (see Read 2006). Cases like this,
which violate only a kind of semantic “principle of sufficient
reason”, were often included under the heading
“insolubles” (for example, Buridan, Sophismata
VIII.8). A variety of epistemic and pragmatic puzzles were often
included as
well.[32]
There is often no attempt, as is usual in present-day literature on
the paradoxes, to ignore all the inessentials and focus in on a single
paradigmatic case that gets at the kernel of the issue. For medieval
authors, the issue was a broad one. Most did not attempt to give any
precise and rigorous characterization of what it takes to be an
insoluble. Often, the definitions they did give are quite general and
include much more than Liar-type paradoxes. In contrast,
Bradwardine’s definition is precise: an insoluble is “a
difficult paralogism secundum quid et simpliciter arising
from some [speech-] act’s reflection on itself with a privative
determination” (Bradwardine [B-I]: §2.1).

Secondly, medieval authors did not have any sense of theoretical
“crisis” over insolubles, as modern discussions of the
paradoxes often do. The medievals did not regard the paradoxes as
threatening the very foundations of reasoning. On the contrary, most
authors seem to have regarded them as merely argumentative nuisances,
and their main concern was to come up with a way of dealing with them
when they arose in disputation. No doubt this difference from the
reaction to the logical paradoxes in modern logic is due to the
different contexts in which the discussions emerged. Modern logic is a
formalized, systematic discipline, closely tied to the foundations of
mathematics; medieval logic, by contrast, was much looser and informal
(which of course is not to say it lacked insight), much more tied to
the give and take of live academic disputation.

Thirdly, and related to the second point, most medieval authors
thought it was entirely feasible to find a completely satisfactory
“solution” to insolubles. Insolubles were regarded as
resting on a straightforward but pernicious fallacy, although authors
disagreed over just what the fallacy is. William of Ockham, for
instance, writes,

As for insolubles, you should know it is not because they can in no
way be solved that some sophisms are called insolubles, but because
they are solved with difficulty. (Ockham [O-SL]: III-3,
46)

The only medieval author who is known to have departed from this
confident view is William Heytesbury, who raises objections against
his own view, and then remarks (Heytesbury [H-OI]: 45, emphasis
added):

Many objections of this sort can be raised against this view, which it
would be difficult or impossible to answer to complete
satisfaction.

Again, about his own view he says (p. 21, emphasis added):

I do not claim that it or any [opinion] is altogether satisfactory,
because I do not see that this is possible. Nevertheless I
rate this one among all of them to be nearer the truth.

Just as the bond of love is sometimes called insoluble, not because it
can in no way be untied (sit solubilis) but because it can be
untied [only] with difficulty, so a proposition is sometimes called
insoluble, not because it is not solvable but because it is solvable
[only] with difficulty.

Anonymous, Treatise on Insolubles, in Peter of Spain:
Tractatus syncategorematum and Selected Anonymous Treatises,
(Mediaeval Philosophical Texts in Translation, vol. 13), Joseph P.
Mullally (trans.) Milwaukee, WI: Marquette University Press, 1964, pp.
335–9. Among the “anonymous treatises” translated at
the end of this volume, this is a late (probably fifteenth century)
treatise on insolubles.

Aristotle, On Interpretation [De
Interpretatione], E.M. Edghill (trans.), in The Works of
Aristotle, Sir David Ross (ed.), Cambridge: Cambridge University
Press, 1928.

Bonaventure, “Quaestiones disputatae de mysterio
Trinitatis”, in Medieval Philosophy: From St. Augustine to
Nicholas of Cusa, (Readings in the history of philosophy), John
F. Wippel and Allan B. Wolter (trans.), New York: Free Press,
1969.

Buridan, John, [B-S], Sophisms on Meaning and Truth,
Theodore Kermit Scott (trans.), New York: Appleton-Century-Crofts,
1966. A translation of Buridan’s Sophismata, based on
the edition published in Buridan [B-S2]. Chap. 8 is on
insolubles.

–––, [B-B], John Buridan on
Self-Reference, G.E. Hughes (ed. and trans.), Cambridge:
Cambridge University Press, 1982. A translation, with philosophical
commentary, of Chap. 8 of Buridan’s Sophismata, on
insolubles. [Note: There are two versions of this book, with different
pagination. The paperbound publication has the subtitle: Chapter
Eight of Buridan’s “Sophismata”, translated with an
Introduction and a philosophical Commentary. The hardbound
publication includes a Latin edition, and has the slightly different
subtitle: Chapter Eight of Buridan’s
“Sophismata”, with a Translation, an Introduction, and a
philosophical Commentary.]

Fland, Robert, Insolubilia, in Paul Vincent Spade, 1978,
“Robert Fland’s Insolubilia: An Edition, with
Comments on the Dating of Fland’s Works”, Mediaeval
Studies, 40: 56–80. Read and Thakkar (2017) argue that the
author’s real name was Robert Eland. He wrote between 1335 and
about 1360. doi:10.1484/J.MS.2.306221

Secondary Literature

Alwishah, Ahmed and David Sanson, 2009, “The Early Arabic
Liar: the Liar Paradox in the Islamic World from the Mid-Ninth to the
Mid-Thirteenth Centuries CE”, Vivarium, 47(1):
97–127. doi:10.1163/156853408X3459090909

–––, 1983, “Roger Swyneshed’s Theory
of Insolubilia: A Study of Some of his Preliminary Semantic
Notions”, in History of Semiotics, (Foundations of
Semiotics, 7), Achim Eschbach and Jürgen Trabant (eds.),
Amsterdam: John Benjamins, pp. 105–114. Reprinted in Spade
1988.

–––, 1988, Lies, Language and Logic in the
Later Middle Ages, (Variorum Collected Studies Series), London:
Routledge. A collection of seventeen previously published papers,
seven of them on insolubles.